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Mademuasel [1]
3 years ago
13

Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.

Mathematics
1 answer:
grandymaker [24]3 years ago
7 0
(2x+1)^{\cot x}=\exp\left(\ln(2x+1)^{\cot x}\right)=\exp\left(\cot x\ln(2x+1)\right)=\exp\left(\dfrac{\ln(2x+1)}{\tan x}\right)

where \exp(x)\equiv e^x.

By continuity of e^x, you have

\displaystyle\lim_{x\to0^+}\exp\left(\dfrac{\ln(2x+1)}{\tan x}\right)=\exp\left(\lim_{x\to0^+}\dfrac{\ln(2x+1)}{\tan x}\right)

As x\to0^+ in the numerator, you approach \ln1=0; in the denominator, you approach \tan0=0. So you have an indeterminate form \dfrac00. Provided the limit indeed exists, L'Hopital's rule can be used.

\displaystyle\exp\left(\lim_{x\to0^+}\dfrac{\ln(2x+1)}{\tan x}\right)=\exp\left(\lim_{x\to0^+}\dfrac{\frac2{2x+1}}{\sec^2x}\right)

Now the numerator approaches \dfrac21=2, while the denominator approaches \sec^20=1, suggesting the limit above is 2. This means

\displaystyle\lim_{x\to0^+}(2x+1)^{\cot x}=\exp(2)=e^2
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A circular sign is precisely designed to have an area of 50.24 square inches. What is the sign's circumference?
choli [55]

Step-by-step explanation:

First of all we need to know the formula for the circumference which is: c=2\pi r

We don't have the radius. What we only have is the area; therefore, we must use the area formula and extract the radius from it.

The formula for the area is: A=\pi r^2 Solve for r;

r^2=\frac{A}{\pi}\\ r=\sqrt[]{\frac{A}{\pi} }

r=\sqrt[]{\frac{50.24inch^2}{3.14} }

r=\sqrt[]{16inch^2}\\ r=4inch

Now that we've found the radius, we simply plug it into the circumference formula.

C=2\pi r\\C=2(3.14)(4)\\C=25.12inch

5 0
3 years ago
Determine the relation of the following lines y = -2x +4 and y= -2x +1
Ivahew [28]
They're parallel because they have the same slope.
7 0
3 years ago
How are a common denominator and a common multiple alike and diferent
marin [14]

Answer:

To find a common denominator of two fractions, you have to find a number that both denominators of the given fractions will divide into. And to find a common multiple, you have to find a number that both given numbers will divide into.

Step-by-step explanation:

6 0
3 years ago
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Cross multiply to get answer<br> x + 5/10 = 2x-3/15
Andreyy89

Once cross multiplied you end up with this:

15x+75=20x−30

Next subtract 20x from both sides.

15x+75−20x=20x−30−20x

−5x+75=−30 -----------> Answer

Now subtract 75 from both sides

−5x+75−75=−30−75

−5x=−105

Now divide by -5 on both sides.

-5/-5x = -105/-5

<h3 /><h3>x = 21 --------------> ANSWER</h3>
3 0
3 years ago
Quadrilateral ABDE is a rectangle. DE = 4 in. and BD = 6 in. If the rectangle is rotated about a line through CF, cylinder Y is
postnew [5]

Answer:

1 in. is the answer.

Step-by-step explanation:

Solution: A Rectangle ABDE in which DE=4 inches and BD= 6 inches

 There are two kinds of rotations

1. one along the Breadth, side having length 6 inches,i.e rotated along line GH,  Cylinder Z is created.

Radius of cylinder Z=6/2= 3 inches

2. Second along the Length, Side having length 4 inches,i.e rotated about a line CF, cylinder Y is created.

Radius of cylinder Y= 4/2= 2 inches

Difference in Radii= Radius of cylinder Z  -  Radius of cylinder Y

                              =  3 - 2= 1 inches

Diagram shown below of both the cases:

7 0
3 years ago
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